Note: This essay was originally written in 2019 or 2020 and stored in a separate “Writing” section of my site, which no longer exists. Since I’ve linked to it in other places and have received a fair bit of correspondance about it, I am turning it into a blog post so it will live forever.
If you attended public school during the last 50 years, your ability to think about and work with numbers and mathematical concepts was likely damaged at a young age - damaged by the negligence of the very system your parents trusted to help you. Maybe you don’t see the harm. Perhaps you think you don’t really need math anyway. This is an illusion. If you would like to keep this illusion and remain in blissful ignorance, I suggest you stop reading this right now. What you’re about to learn can’t be unlearned, and once you have read it, you will never see it the same way again.
What Is Math Abuse?
Imagine if your high school had allowed you to graduate without being able to read. How could you function as a normal adult? You would have been sentenced to a life of difficulty.
In fact, it is quite likely that such sabotage was actually performed on you. However, it was not your literacy, but your numeracy that was attacked. This is not a screed against the state of public schooling (I will not call it “education”) but against a specific injury it tends to inflict on those unfortunate enough to be left in its “care” for 12 years. I refer to a tragic phenomenon that I call “Mathematical Child Abuse,” or “Math Abuse” for short.
If you are like most people, the very word “Mathematics” has been poisoned for you from the beginning. It likely conjures an image of a stuffy elementary school classroom where a strict teacher forces you to recite the dreaded “times tables” from memory while you watch through the window as another beautiful spring day passes away forever. If you somehow escaped elementary school math without incident, then you might have first felt the pain in middle school, where the infamous “Pre-Algebra” courses drilled yet more useless rote memory skills into you in preparation for some mystery known as “Algebra,”. Maybe you were an unusual kid, though. Perhaps you were a “math person” (more on that little fiction later) and got along quite well in “Pre-Algebra.” Surely, then, your hopes must have been finally dashed in high school, where you came face to face with the ravenous beast that has devoured entire generations’ innocence and thirst for knowledge. I refer, of course, to the Standard Mathematical Curriculum.
The Standard Mathematical Curriculum
The SMC is the main vehicle for Math Abuse. Rather than describe it myself, I will defer to Paul Lockhart, who explains it much better (and funnier) than I ever could:
The Standard School Mathematics Curriculum
LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.
MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison. Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’ and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent preparation for Algebra I.
ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.
GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.
ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.
TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues.
CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim.
If you laughed or cried while reading the above, you had good reason. This soul-crushing mountain of garbage is literally the same in every public school in the United States. It’s perfect for corporate lobbyists textbook companies and associated grifters educators, because it’s standardized to such an extent that students’ “progress” can be predicted and their work planned years in advance, reducing still further the amount of real work anyone in the whole craven system must actually do. If you want to train the next generation of employees to sit quietly for hours and do busywork, this is the way to do it. The problem is, this approach also leads to certain “anxieties,” “disorders,” and “phobias” as well…
In about the past 20 years, Attention Deficit Disorder (ADD), Attention Deficit Hyperactive Disorder (ADHD), and various terms such as “math anxiety” have entered the public lexicon. I will not make this a rant about how ADD/ADHD are massively over-diagnosed. I will also not rant that our schools seem all too ready to dispense powerfully mind-altering prescription amphetamines to young boys if they commit the crime of wanting to play outside rather than sit unnaturally still for 8 hours (Not that I’m bitter from personal experience or anything.) No, I’ll save the modern mania of pathologizing normal behavior for another article. For the present article, I’d merely like to suggest that while these maladies are thought to be unfortunate hurdles that make school hard for certain kids, the truth is that school itself causes these symptoms! Plenty of kids are stressed about English, gym, or socialization, but I would suggest from personal experience (and I’m sure many will agree) that nothing gets you stressed in K-12 like math.
People have been talking about “math anxiety” and “test anxiety” for a while now.
The Myth of the “Math Person”
We’re now well on the way to demonstrating that “math anxiety,” “test anxiety,” and probably a big chunk of alleged diagnosed “behavior disorders” are actually caused by school. I now intend to disprove the oft-quoted fallacy that there exist “math people” who are naturally predisposed to be good at math, and “other people” who are not. If you’ll bear with me, I’ll do this by example of my own personal journey. Feel free to skip past if you’re not interested in autobiographical details.
Aside: Personal Anecdote on Becoming a “Math Person”
My Earliest Intuition
At the top of my classroom’s walls in kindergarten, there was a long, colorful paper sheet with the numbers 1 - 100 on it. One day I noticed that the right-most numbers kept going “1, 2, 3, 4, 5 … 9” over and over, even when they were part of bigger numbers like “24” or “81”. Furthermore, the numbers on the left also went “1, 2, 3, 4, 5 … 9”, but slower than the ones on the right. The numbers didn’t go past 100, but I had a hunch that the next one would be “111”. (I was 5, give me a break.) The point I’m trying to make is that I came very close to understanding the positional number system. Too bad that I was in public school and already in the grip of the SMC, otherwise someone may have been prepared to explain things to me ahead of “grade level” (the horror!) and introduce me to truths I would sadly not learn until years later.
I Encounter the Mystery
I remember sitting in my 3rd grade classroom and hearing the teacher’s droning voice fade into a dull background noise. While she repeated the same points on the same topics from the same lessons for days in a row lest some Child be Left Behind, I would absentmindedly flip through my textbook. I wouldn’t read the part we were on, though: I’d flip to the end, a mysterious region we somehow never reached. That chapter was devoted to angles, a subject I had not yet been formally introduced to. I knew what a “right angle” was, that it was “90 degrees,” and that it was the angle made by a perfectly square corner, but that was all. The pages at the end of the book contained diagrams of circles, chords, tangent lines, and arcs. When I noticed them for the first time, they got some kind of hold on me. They were puzzling and wonderful, mysterious yet aesthetically pleasing in form. I kept turning back to them, not because I understood them (I wouldn’t get the chance in that class), but because they were beautiful. I didn’t know it then, but I had just had my first aesthetic experience with mathematics.
Struggle and Triumph
After that experience, I became obsessed with the trappings of “advanced math.” I would doodle things like “√x2” in the margins of my notebook. I began to amass knowledge of certain trivia, like the fact that “√” meant “square root,” that “2” meant “squared,” that “3” meant “cubed,” and so on. I could write “E = mc2”, and I knew that “E” meant “energy,” “m” meant “mass,” and “c2” meant “the speed of light squared,” but I didn’t know what any of that really meant. I can see now that I was trying to recreate the aesthetic experience of seeing the angles for the first time. Mathematical notation was sort of a mysterious alchemical code, legible only to its wise initiates. I wanted to be one.
Sadly, I believed that I never could be. Why should I aspire to be someone who understood “E = mc2” if I couldn’t even complete a multiplication worksheet correctly in under 60 seconds? What hope did I have of understanding those mysterious symbols if I couldn’t even divide 38 into 743 before the teacher put the next problem on the overhead? How would I ever be a scientist or an engineer or an astronaut if I couldn’t immediately identify the greatest common factor or the lowest common divisor of some arbitrary number?
In hindsight, it’s easy for me to see that earning a B.S. in aerospace engineering with a concentration in astronautics did not, in fact, require performing such stupid little parlor tricks that seemed all so important in the fifth grade. What held me back all those years was the myth of the “Math Person” and my despondent belief that I was not one. I was in college before I managed to truly undo the psychological harm left by K-12 math. My first math course in my college career was Calculus I, which I finished with a “C” average. A scant two years later, I finished Calclulus IV with an “A.”
Over the four years of undergrad, I also managed to complete Calculus II, III, and IV, physics I, II, and III, orbital mechanics I and II, differential equations, linear algebra, aerothermodynamics, compressible fluid mechanics, spacecraft attitude control dynamics, and much, much more. I had finally attained my goal: to be one of those initiated wise men who understood all those strange and wonderful symbols. It was as though I had learned to speak another language.
What was it that caused such a sea change in my approach to mathematics? It all came down to the fact that I was really interested in aviation, specifically in spaceflight. I knew that I wanted to be part of that somehow. I knew also that there would be a lot of science and math involved. So, I chose a major and a career path that would basically force me to confront it. I supposed that if I really wanted it bad enough, I could figure it out as I went. I was right!
The Road to Healing
Maybe I should have said this near the top: Please don’t take the tone or content of this article as making light of abuse or of co-opting terms used for people who are recovering from abuse. I’m using the term “Math Abuse” and speaking of “recovery” because that’s pretty much the only phrase I can think of that comes close to the seriousness of this topic. As I alluded to in the beginning, we place a high value on literacy, but very little public praise is given to numeracy.2 If the tables were turned and people graduated high school with reading skills as stunted as their math skills (I know what you’re thinking, but we’re not quite there yet) there would be public outcry and people would indeed talk about “abuse” and the need to “recover” this most precious skill. It frustrates me that few people are doing this now.
Learn Math Physically!
I can remember a lot of misguided attempts to cater to certain “learning styles” when I was in school.3 I don’t believe there are “visual people” or “auditory people” or “kinesthetic people” (except maybe those with true cognitive disabilities), but I do think certain presentation methods are suited for particular subjects. For this reason I advise you to do a few physical things when learning math as an adult:
Use paper and pencil, NOT a calculator, computer, or phone. Doing this forces you to be deliberate and to work no faster than you can think. By slowing down, you avoid the muddle-headedness that comes from trying to think so quickly that you get ahead of yourself.
Use visual aids … a lot! This goes without saying for geometry, but it’s helpful for many other things. When considering the behavior of a function, graph it! (On paper, of course.) If you can’t tell where the value of a trigonometric function lies, sketch a unit circle. (If you get creative, you can solve unlikely-seeming problems with pictures. During the SAT, I had to compute the tangent of an angle, but my calculator didn’t have trig functions. I improvised by drawing the angle with a protractor, turning it into a right triangle with a straightedge, and dividing the length of the opposite side by the length of the adjacent side, then selecting the closest answer.)
Consider going old school (slide rules, abaci, etc.) This isn’t just for hipsters. You’d be surprised what intuitions you’ll have if you calculate things the old way. If you get proficient at using slide rules (it doesn’t take long) you’ll naturally get proficient at estimating orders of magnitude and develop some nice mental arithmetic skills in the bargain. The abacus may not provide the same level of insight, but the mystery of positional numbers is more obvious when using one (even most adults don’t fully get it.) You don’t even need a purpose-built abacus; you can make an impromptu one by scratching rows in the dirt and placing pebbles in them.4
Read … A Lot
Isaac Asimov, one of the “Big Three” science fiction writers (along with Robert Heinlein and Arthur C. Clarke) and one of the most prolific writers of the twentieth century to boot, published fiction and nonfiction books in every single category of the Dewey Decimal Classification except for category 100 (philosophy and psychology.) Of all these, some of the most valuable to me personally were his “Realm of …” series, such as Realm of Algebra, Realm of Numbers, and so on. I fully believe you could replace the entire K-9 curriculum with just those two titles. I discovered them in college, and my first reaction was something like “Why didn’t anyone tell me these existed?!” As an example of just how clearly and succinctly he managed to explain things that took my K-12 teachers and textbooks days of waffling to imperfectly communicate, I’ll quote from Realm of Numbers on the subject of imaginary numbers:
In short, if we only knew the square root of minus one (√-1) we could work out the square root of any negative number. But here we are faced with the dilemma already mentioned: 1 × 1 = 1 and −1 × −1 = 1. No number multiplied by itself will give -1.
The only thing we can do is invent an answer.
We could, for instance, define a sign such as # and say that # × # gives a negative number. Then #1 × #1 = −1. That would be true by definition and if it didn’t contradict anything else already established in the mathematical scheme, there would be no reason why we couldn’t get away with this.
Of course, such a number would seem to be an imaginary one. We know what +$1 and −$1 are. The first is a one-dollar asset and the latter a one-dollar debt. What would #$1 be? The first mathematicians who worked with the square roots of negative numbers called them “imaginary numbers” and the name persists to this day. Ordinary numbers, whether positive or negative, rational or irrational, are called “real numbers” in contradistinction.
The mathematicians have not made up a new sign, however, corresponding to + or −. (I wish they had; it would have been neater.) Instead, they invented the symbol i (for “imaginary”) and let that stand for the square root of −1.
However, you may, by now, be bursting with indignation. Never mind all this talk about i, you may be thinking. What does imean?
Actually, it can mean whatever we choose to make it mean. You must remember that numbers are only a creation of man to help him understand the universe about him. We can do what we want with our own creations as long as it helps us understand.
[ed: Asimov then goes on to describe how the number line containing positive and negative numbers might be expanded to become a number area, adding a perpendicular axis for the imaginary numbers. This is the complex plane, and all numbers may be thought of as points on this plane with both a real and an imaginary component. This took me until my sophomore year of college to understand. If I had read this book in high school, I would have understood in an instant.]
Isn’t that great? When I first read that passage, it only further cemented in my mind the suspicion that the people who write the textbooks and curricula in this country could be doing things a lot better, but for some nefarious reason or other, they don’t.
^ I detest their methods, but you’ve got to admire their ROI. If I had no morals, the textbook/curriculum industry would be a pretty sweet gig.
^ In fact, quite to the contrary, many people seem to make a sport of competing to see who has the least aptitude for math. I can understand a little self-deprecating humor, but when people chuckle about how things are “too technical” for them or that they “don’t have that kind of mind,” I think it’s sad. They’ve given up; they’re beaten before they’ve tried.
^ Another modern fad, like the ADD/ADHD mania, in which I am reluctant to believe. I happen to think that certain topics lend themselves to particular presentation styles, e.g. music is necessarily auditory, much of physics is necessarily visual, etc. but I remain unconvinced that people have any natural inclination to particular modes of learning. This is a commonly-heard excuse from the type of person who says something is “too technical” or “my brain doesn’t work that way” when really he’s just refusing to try.
^ An ancient Roman would likely have done this if he didn’t have access to a fancy rod-and-bead abacus. The latin word meaning “small stone” is calculus. Consequently, to this very day, when handling numbers we say that we are “calculating.”
^A Mathematician’s Lament (also called Lockhart’s Lament) by Paul Lockhart (2002). PDF (391K)